\section{Partial Volume Pricing market}
\label{sec:volume}

\subsection{Market Model}

We also consider a partial volume pricing market model. The market is
slightly different from the flat pricing market in that the provider
should decide on the volume-based price, $p_v^M,$ when a user is
served by a macro BS, and a user should also decide on the elastic
data demand $x^M.$ In this section, we only consider the case of the
\emph{open-to-femto} policy for the following reason. When $p_m = 0,$
in the \emph{open-to-all} policy, \emph{mobile-only} users can use a
free \emph{open-femto} service, in which case the provider's revenue
is significantly reduced because of free-riding.

The provider selects the optimal prices that
maximize the following problem:
\begin{eqnarray} 
{\bf Provider:} & \max_{p_v^M,p_{o},p_{c}}&R \cr 
& \text{s.t} &
  p_v^M,p_{o},p_{c} \ge 0,
\end{eqnarray}

Then, a user with type $\gamma$ first determines the data demand for
macro BSs $x^{M}(\gamma)$ by maximizing the corresponding surplus
subject to the macro BS capacity constraint. He or She then selects a
service type to maximize the net-utility.
\begin{eqnarray}
\label{eq:user}
{\bf User:} && x^{M}(\gamma) =\arg \max_{x }\gamma x^{\theta} -
p_{v}^M x, \cr
&& l^*(\gamma) =\arg \max_{l \in \{ m,o,c \}} \tilde{U}_{l}(\bm{x};\gamma).
\end{eqnarray} 

The total amount of traffic must be less than the capacity of the
macro BS. Thus, every user can be served according to his or her entire demand
$x^{M}(\gamma)$, when the following condition is satisfied. 
\begin{equation}
T^{M}\triangleq N\int\pi_{l^*(\gamma)}^{M}x^{M}(\gamma)d\gamma\le C_{M},\end{equation}
where $T^M$ denotes the total macro BS traffic generated by
users. When $T^M$ exceeds $C_M$, as in flat pricing, we assume that
the fair scheduler controls the serving rate. Thus, the service rate
is suppressed by an upper bound $x^M_{max}$, where  
\begin{equation}
T^{M} = N\int\pi_{l^*(\gamma)}^{M} \min \{ x^{M}_{max},
x^{M}(\gamma) \}d\gamma = C_{M}.\end{equation}
Note that unlike flat pricing where some users exit from the
market and subscribe to no services,
every user selects one of the services in partial volume
pricing. The revenue of the operator simply reads:
\begin{equation}
R=T^M p_{v}^M+N\big\{(p_{o}-c_{f})\alpha_{o}+(p_{c}-c_{f})\alpha_{c}\big\}.\end{equation}


\subsection{Equilibrium}
\label{sec:partial_eq}

Computing the equilibrium in partial volume pricing is even harder
than it is in flat pricing. This difficulty is caused by the hybrid
structures of the two pricing schemes, where the volume pricing for
macro BSs often causes the net-utility to be non-linear, as shown in
Fig.~\ref{fig:flat_price}(b). When the net-utility graph has only one
intersection point between any two lines, it is relatively easy to
find the equilibrium point (as in the flat pricing case). However,
this non-linear net-utility graph sometimes generates multiple
intersections between any two net-utility curves (e.g., the bottom of
Fig.~\ref{fig:flat_price}(b)), leading to difficulties in finding the
relationship between $\bm{\alpha}$ and $\gamma,$ which is the first
step in computing the equilibrium.

For simplicity, we consider the case when $\bar{\gamma}=1$. Similarly
to flat pricing, we develop a theorem to compute the equilibrium.  We
again use the notations $x^L(\bm{\alpha}),$ $L \in \{O,C\}$ and
$x^M(\bm{\alpha};\gamma)$ to explicitly show the dependence of the
data rates on $\bm{\alpha}.$ We use $\bm{x}(\bm{\alpha})$ to denote
the vector of $x^L(\bm{\alpha}),$ and omit $\gamma$ for notational
simplicity, unless required. 

In the description of Theorem~\ref{thm:pvolume}, similarly to
Theorem~\ref{thm:palpha}, we use the index variables $i,j.$ When
$x^{O}(\bm{\alpha})< x^{C}(\bm{\alpha})$, $i=\singleq{o},$ $j=\singleq{c}$, otherwise,
$i=\singleq{c},$ $j=\singleq{o}$.

% In the description of Theorem~\ref{thm:pvolume}, similarly to
% Theorem~\ref{thm:palpha}, we use the index variables $I,J,i,j.$ When
% $x^{O}(\bm{\alpha})> x^{C}(\bm{\alpha})$, $I=\singleq{O},$
% $J=\singleq{C}$ (and $i=\singleq{o},$ $j=\singleq{c}$), otherwise,
% $I=\singleq{C},$ $J=\singleq{O}$ (and $i=\singleq{c},$ $j=\singleq{o}$).
\begin{theorem}
  \label{thm:pvolume} We define $\set{A}$ 
  \begin{equation} \set{A} \triangleq \{ \bm{\alpha} \mid
    x^{M}(\bm{\alpha};\gamma) \leq \min \{ x^{O}(\bm{\alpha}) ,
    x^{C}(\bm{\alpha}) \}, \text{for all } \gamma \in (0,1]
    \}. \end{equation} 
  % We will use $I=\singleq{O}$ (resp. $i=\singleq{o}$) and
  % $J=\singleq{C}$ (resp. $j=\singleq{c}$), if $x^{O}(\bm{\alpha})>
  % x^{C}(\bm{\alpha})$. Otherwise, we will use $I=\singleq{C}$ (resp. $i=\singleq{c}$) and
  % $J=\singleq{O}$ (resp. $j=\singleq{o}$). 
 % Let $i$, $I$, $j$, and $J$ indicate $o$, $O$, $c$, and $C,$ respectively if $x^{O}(\bm{\alpha})>
 %  x^{C}(\bm{\alpha})$, and otherwise $i$, $I$, $j$, and $J$ refer to
 %  $c$, $C$, $o$, and $O,$ respectively. 

  Then, for all
  $\gamma \in (0,1]$ and any given $\bm{\alpha} \in \set{A},$
  \begin{compactenum}[(i)]
  \item 
    The $\gamma_{mi}$ and $\gamma_{ij}$ are unique and given by:
   \begin{eqnarray}
      \gamma_{mi} = 1-\alpha_i-\alpha_j, \quad \gamma_{ij} = 1-\alpha_j.
    \end{eqnarray}
  \item The $p_{i}$ and $p_{j}$ are then expressed as a closed form of
    $\bm{\alpha}$ in the following manner:
    \begin{eqnarray}
      p_{i}&=&U_{i}({\bm x};\gamma_{mi})-U_{m}({\bm x};\gamma_{mi})+p_{v}^{M}x^{M}(\bm{\alpha};\gamma_{mi}),\cr
      p_{j}&=&p_{i}+U_{j}({\bm x};\gamma_{ij})-U_{i}({\bm x};\gamma_{ij}).
    \end{eqnarray}
  \item  $p_v^M = \infty$ maximizes the provider's revenue if 
    \begin{equation} \label{con:p}
      \frac{\theta}{2-\theta}\frac{(1-\pi_{i}^{M})(\gamma_{mi})^{\frac{2-\theta}{1-\theta}}+\pi_{i}^{M}}{(1-\pi_{i}^{M}) (\gamma_{mi})^{\frac{1}{1-\theta}}(1-\gamma_{mi})}<1.
    \end{equation}
    Otherwise, the following $p_v^M$ maximizes the revenue:
 %   \sqeq
    \begin{equation}
      \label{eq:lbound}
      p_{v}^{M}=\theta\Bigg(\Big(\frac{1-\theta}{2-\theta}\Big)\Big((1-\pi_{i}^{M})\gamma_{mi}^{\frac{2-\theta}{1-\theta}}+\pi_{i}^{M}\Big)\Big(\frac{N}{C_{M}}\Big)\Bigg)^{1-\theta}.
    \end{equation}
%    \unsqeq
  \end{compactenum}
\end{theorem}
\medskip

The proof is presented in the Appendix. Note that $\set{A}$ contains
all $\bm{\alpha}$ where femtocells give better throughput to users
than that of macrocells. Therefore, $\set{A}$ includes all cases where
femtocells generate additional revenue to the operator.

% Similar to $\set{A}$ in flat
% pricing, the set $\set{B}(\gamma)$ is highly comprehensive and includes
% most practical situations. In general, there are many users in each
% cell. Thus, since every user has a certain degree of data demand for the macro BS,
% $x^M(\gamma )$ should be much smaller than $C_M$ for all user type
% $\gamma$.

% Moreover, in practical system, $C_F > C_M$. Therefore, in most of practical cases, $x^M \le x^O.$

Theorem~\ref{thm:pvolume}(i) states that a relationship
between the subscription ratio and the type $\gamma$ can be
simply characterized.
Theorem~\ref{thm:pvolume}(ii) represents $p_o$ and $p_c$ as
 simple functions of $\bm{\alpha}.$ In Theorem~\ref{thm:pvolume}(iii),
the equations (\ref{con:p}) and (\ref{eq:lbound}) are the functions
of $\bm{\alpha},$ for example, $\gamma_{mi}$ and $\pi_i^M$ are 
determined for a given $\bm{\alpha}.$ Thus, a simple optimization can be
used to compute the equilibrium as in Theorem~\ref{thm:palpha}.

In Theorem~\ref{thm:pvolume}(iii), the left-hand side of the condition
(\ref{con:p}) is more likely to be satisfied when there are more femto
users, because $\gamma_{mi}$ declines for such a case. Thus, it
means that with many femto users, $p_v^M$ should be large to increase
revenue, because with small $p_v^M$, more users tend to subscribe to
the {\em mobile-only} service. In such a case, the provider will
decrease $p_o$ and $p_c$ to attract more femto users, resulting in an
overall decrease in revenue. Note that for a small
number of femto
users (i.e., (\ref{con:p}) is violated), the provider gets more revenue
with smaller $p_v^M,$ wherein the data demand for macro BSs will grow. 
Thus, in order to maximize revenue, $p_v^M$ is decreased to (\ref{eq:lbound}) until the data
demand at macro BSs reaches its capacity. Note that even when $p_v^M$ is
very large, where users decrease the traffic demand for macro BSs,
$\gamma_{mi}$ can be some positive value. 




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